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Mike Keith's website uses a shorthand notation for these numbers. The 4-primeval prime 100111222333444555666777998889 is written in this notation as (1) 2 3 3 3 3 3 3 3 0 998889. The (1) represents the leading 1 digit (which will always be present). The next number says how many consecutive 0's follow the leading 1, and the next says how many consecutive 1's follow that, and so on up to the number of consecutive 8's. The final grouping explicitly shows how the last group of 8's and 9's are arranged.
The following are the n-primeval primes as found by Jerome Storti in this notation:
5 (1) 3 3 4 4 4 3 3 4 0 98889899
6 (1) 4 4 4 5 5 5 5 5 4 999989
7 (1) 5 5 5 6 5 5 5 6 3 98899999
8 (1) 5 6 7 7 6 7 7 7 6 98999999
9 (1) 7 7 8 8 8 7 8 8 6 9999989899
10 (1) 8 8 8 9 9 9 9 9 7 9999899999
11 (1) 8 9 10 10 10 9 10 10 6 9889989999999
12 (1) 10 10 10 11 11 11 10 11 9 9998999999899
13 (1) 10 11 11 12 11 12 11 12 9 99899999999899
14 (1) 11 13 13 13 12 12 12 13 11 989999989999999
15 (1) 12 13 14 14 13 14 13 14 12 9999999988999999
16 (1) 13 14 14 15 14 14 14 15 12 99999999999999889
17 (1) 14 15 15 16 15 15 15 16 14 998999999999998999
18 (1) 16 17 17 17 16 17 17 17 14 9989999999999899999
19 (1) 17 18 17 18 17 17 17 18 15 988999999899999999999
20 (1) 17 19 18 19 19 18 19 19 16 999999998999999999989
21 (1) 18 19 19 20 19 19 20 20 17 9899999999999999998999
22 (1) 18 20 20 21 20 21 21 21 18 99998999999999999998999
23 (1) 21 23 21 22 21 21 22 22 19 999999889999999999999999
24 (1) 20 22 22 23 22 22 22 23 21 999999999999999989999999
25 (1) 23 23 23 24 23 23 23 24 22 9999999999999999998999999
26 (1) 23 24 24 25 25 25 24 25 22 999999999999999999899999989
27 (1) 24 25 25 26 25 25 25 26 23 9999999998999999999999998999
28 (1) 25 26 27 27 27 26 27 27 25 9999899999999999999999999999
29 (1) 25 27 27 28 27 27 27 28 25 999999989999999999999999999989
30 (1) 26 29 28 29 29 28 28 29 27 999999999999998999999999999999
31 (1) 28 29 29 30 29 29 29 30 27 99999889999999999999999999999999
a(2) = 1123465789 because this is the smallest prime out of which each of the first 25 primes below 10^2, viz. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 can be formed using its digits.
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